conditional probability

Introduction to Conditional Probability

• Definition: Conditional probability is essential in statistics because it allows us to understand how the occurrence of one event affects the probability of another associated event.

Importance of Conditional Probability

• When two events are linked, knowing one event alters the probabilities associated with the other.

• Example:

  • Probability of Rain: Suppose the probability of rain in an area is stated as 40%. This figure is not arbitrary but contingent upon various related factors, such as the approach of a cold front or the presence of rain clouds.

  • If one of these factors changes, the probability of rain will also be affected, hence making it a conditional probability.

Example of Dice Rolls

• Basic Probability: When throwing two fair dice, the probability that the dice sum to three is calculated based on equally likely outcomes.

  • Total Outcomes: There are 36 possible outcomes when rolling two dice.

  • Favorable Outcomes: There are only two outcomes that yield a sum of three: (1,2) and (2,1).

  • Therefore, the probability of the event (A) is:
    $P(A) = \frac{2}{36} = \frac{1}{18}$

• Modified Experiment:

  • If we first roll the white die and it shows 1, the question arises: what is the probability that the two dice sum to three?

  • Event Definitions:

  • Event A: Sum of dice is three.

  • Event B: White die shows one.

  • New Sample Space: Knowing the outcome of the white die reduces the sample space to just six possible outcomes (the black die can be 1, 2, 3, 4, 5, or 6).

  • The only combination that sums to three, given that the white die is one, is (1,2).

  • Therefore, the new probability is:
    $P(A|B) = \frac{1}{6}$

  • This indicates that given the condition that the white die is one, our understanding of the event (dice summing to three) has changed markedly.

Association Between Events

• Titanic Example:

  • The assumption is made that first-class passengers had a higher survival probability compared to second or third class passengers.

  • Probabilities:

  • Probability of first-class survivors:
    $P(\text{First Class and Survived}) = \frac{0.15}{1}$

  • Probability of third-class survivors:
    $P(\text{Third Class and Survived}) = \frac{0.14}{1}$

  • While the probabilities appear similar when viewed through the entire dataset, it's more insightful to focus on the survival rates within each class to determine true conditional probabilities.

Conditional Probability Calculations

• New Perspective:

  • Rather than simply noting passengers' survival in different classes, we should focus on those classified by first or third class.

  • Given a focus on first-class passengers, the survival rate is:
    $P(\text{Survival | First Class}) = \frac{201}{324} \approx 0.62$

  • Conversely, for third-class passengers:
    $P(\text{Survival | Third Class}) = \frac{181}{709} \approx 0.26$

• The marked difference here shows a clearer disparity in survival chances when conditioned on class.

Formal Definition of Conditional Probability

• Phrasing: Conditional probability is framed as "given that" one event has occurred. It refers to the calculation of the likelihood of event A happening, conditioned on event B occurring.

  • The formula for conditional probability is:
    $P(A|B) = \frac{P(A \cap B)}{P(B)}$

• Sample Space Reduction: When event B has occurred, the sample space reduces to only those outcomes that belong in B, allowing the recalculation of the probability for event A based solely on the outcomes in B.

Examples of Conditional Probability

1. Weather Example: Find the probability of rain given that clouds are forming.

2. Dice Example: Calculate probabilities like summing to three based on prior knowledge of outcomes.

Multiplication Rule for Conditional Probability

• The multiplication rule is derived from the definition:
  $P(A \cap B) = P(A|B) imes P(B)$

• If the events are reversed:
  $P(B \cap A) = P(B|A) imes P(A)$

• Important notes:

  • $P(A|A) = 1$ (certainty that event A occurs).

  • For disjoint events A and B, $P(A|B) = 0$ since they cannot occur simultaneously.

Independent Events

• Two events are independent if knowing the outcome of one does not affect the probability of the other:

  • Thus, for independent events, the multiplication rule simplifies to:
    $P(A \cap B) = P(A) imes P(B)$

• Coin Toss Example: Consecutive coin flips do not influence outcomes; thus, probabilities remain 0.5 for heads or tails regardless of past results.

Applying Conditional Probability

• Deck of Cards Example: Calculate probabilities based on known constraints (e.g., a known red card).

• In the scenario where probabilities are calculated given some knowledge, the probabilities will often differ from unconditional probabilities.

Simulation to Understand Empirical Probability

• Definition: Empirical probabilities stem from real-life observations or random actions.

• Importance: They allow confirmation of theoretical probabilities and provide insights when theoretical calculations are impractical.

Simulations in Practice

Example of Coin Flips

• Trial Design: Simulating coin flips or dice rolls helps in estimating probabilities by mimicking real events.

• Application: Simulations can be quickly run through computer software to test hypotheses about probabilities.

• Statistical Findings: Results from simulations over a large number of trials tend to converge towards theoretical probabilities, confirming the law of large numbers.

Conclusion

• Law of Large Numbers: The empirical probability of an event will converge towards the expected theoretical probability as the number of trials increases.

• Gambler’s Fallacy: This highlights common misconceptions regarding independent events and their influence on future outcomes; the belief that past results impact future independent trials.